Question

Consider a game in which, simultaneously, player 1 selects any real number x and player 2 selects any real number y. The payoffs are given by:

u1 (x, y) = 2x − x2 + 2xy

u2 (x, y) = 10y − 2xy − y2.

(a) Calculate and graph each player’s best-response function as a function of the opposing player’s pure strategy.

(b) Find and report the Nash equilibria of the game.

(c) Determine the rationalizable strategy profiles for this game.

Answer 1

In this game there are two players and the “pay-off” function of the each player are also given below.

=> “u1 = 2*X – X^2 + 2*X*Y” and “u2 = 10*Y – Y^2 – 2*X*Y”

=> the best response function of both the player is given below.

=> ?u1/?X = 0, => 2 – 2*X + 2*Y = 0, => X = 1 + Y, be the “best response function” of “playe1”.

=> ?u2/?Y = 0, => 10 – 2*Y – 2*X = 0, => Y = 5 – X, be the “best response function” of “playe1”.

Now, consider the following fig of “best response functions” below.

b).

Now, the “NE” will be determined by the intersection of the “best response” strategies. So, here we have to solve the “best response 1” and “best response 2” simultaneously. So, we can see “E” be the equilibrium here where “best response 1” and “best response 2” intersect to each other. So, the NE is given by “(X, Y) = (3, 2)”.

(Just put the “X” form “best response 1” into “best response 2”, we will get optimum “Y” from “best response 2”, then using “Y” we can find out optimum “X”).

c).

So, here the “rationalizable” strategy profile is given by”(X, Y) = (3, 2)”, => maximize the “pay off” functions.